Question

fully simplify the expression below and write your answer as a single fraction.
$\frac{4x^{5}-100x^{3}}{x^{2}+3x}cdot\frac{x^{2}-9}{4x^{2}-32x + 60}$

Explanation:

Step1: Factor the expressions

Factor $4x^{5}-100x^{3}=4x^{3}(x^{2} - 25)=4x^{3}(x + 5)(x - 5)$; $x^{2}+3x=x(x + 3)$; $x^{2}-9=(x + 3)(x - 3)$; $4x^{2}-32x + 60=4(x^{2}-8x + 15)=4(x - 3)(x - 5)$.

Step2: Rewrite the original expression

The original expression $\frac{4x^{5}-100x^{3}}{x^{2}+3x}\cdot\frac{x^{2}-9}{4x^{2}-32x + 60}$ becomes $\frac{4x^{3}(x + 5)(x - 5)}{x(x + 3)}\cdot\frac{(x + 3)(x - 3)}{4(x - 3)(x - 5)}$.

Step3: Cancel out the common factors

Cancel out the common factors: $4$, $(x + 3)$, $(x - 3)$, $(x - 5)$ and $x$.
We get $\frac{4x^{3}(x + 5)(x - 5)(x + 3)(x - 3)}{x(x + 3)\cdot4(x - 3)(x - 5)}=x^{2}(x + 5)$.

Answer:

$x^{2}(x + 5)$